# An Improvement to the Achievement of the Griesmer Bound

• Volume: 4, Issue: 3, page 301-320
• ISSN: 1312-6555

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## Abstract

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We denoted by nq(k, d), the smallest value of n for which an [n, k, d]q code exists for given q, k, d. Since nq(k, d) = gq(k, d) for all d ≥ dk + 1 for q ≥ k ≥ 3, it is a natural question whether the Griesmer bound is attained or not for d = dk , where gq(k, d) = ∑[d/q^i], i=0,...,k-1, dk = (k − 2)q^(k−1) − (k − 1)q^(k−2). It was shown by Dodunekov [2] and Maruta [9], [10] that there is no [gq(k, dk ), k, dk ]q code for q ≥ k, k = 3, 4, 5 and for q ≥ 2k − 3, k ≥ 6. The purpose of this paper is to determine nq(k, d) for d = dk as nq(k, d) = gq(k, d) + 1 for q ≥ k with 3 ≤ k ≤ 8 except for (k, q) = (7, 7), (8, 8), (8, 9).* This research was partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 20540129.

## How to cite

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Hamada, Noboru, and Maruta, Tatsuya. "An Improvement to the Achievement of the Griesmer Bound." Serdica Journal of Computing 4.3 (2010): 301-320. <http://eudml.org/doc/11390>.

abstract = {We denoted by nq(k, d), the smallest value of n for which an [n, k, d]q code exists for given q, k, d. Since nq(k, d) = gq(k, d) for all d ≥ dk + 1 for q ≥ k ≥ 3, it is a natural question whether the Griesmer bound is attained or not for d = dk , where gq(k, d) = ∑[d/q^i], i=0,...,k-1, dk = (k − 2)q^(k−1) − (k − 1)q^(k−2). It was shown by Dodunekov [2] and Maruta [9], [10] that there is no [gq(k, dk ), k, dk ]q code for q ≥ k, k = 3, 4, 5 and for q ≥ 2k − 3, k ≥ 6. The purpose of this paper is to determine nq(k, d) for d = dk as nq(k, d) = gq(k, d) + 1 for q ≥ k with 3 ≤ k ≤ 8 except for (k, q) = (7, 7), (8, 8), (8, 9).* This research was partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 20540129.},
author = {Hamada, Noboru, Maruta, Tatsuya},
journal = {Serdica Journal of Computing},
keywords = {Linear Codes; Griesmer Bound; Projective Geometry; linear codes; Griesmer bound; generator matrix},
language = {eng},
number = {3},
pages = {301-320},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {An Improvement to the Achievement of the Griesmer Bound},
url = {http://eudml.org/doc/11390},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Maruta, Tatsuya
TI - An Improvement to the Achievement of the Griesmer Bound
JO - Serdica Journal of Computing
PY - 2010
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 4
IS - 3
SP - 301
EP - 320
AB - We denoted by nq(k, d), the smallest value of n for which an [n, k, d]q code exists for given q, k, d. Since nq(k, d) = gq(k, d) for all d ≥ dk + 1 for q ≥ k ≥ 3, it is a natural question whether the Griesmer bound is attained or not for d = dk , where gq(k, d) = ∑[d/q^i], i=0,...,k-1, dk = (k − 2)q^(k−1) − (k − 1)q^(k−2). It was shown by Dodunekov [2] and Maruta [9], [10] that there is no [gq(k, dk ), k, dk ]q code for q ≥ k, k = 3, 4, 5 and for q ≥ 2k − 3, k ≥ 6. The purpose of this paper is to determine nq(k, d) for d = dk as nq(k, d) = gq(k, d) + 1 for q ≥ k with 3 ≤ k ≤ 8 except for (k, q) = (7, 7), (8, 8), (8, 9).* This research was partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 20540129.
LA - eng
KW - Linear Codes; Griesmer Bound; Projective Geometry; linear codes; Griesmer bound; generator matrix
UR - http://eudml.org/doc/11390
ER -

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